Zafar Ahmed

For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time-reversal: \(R(-k)\ne R(k)$ and $T(-k) \ne T(k)\), unless the potentials are real or PT-symmetric. For complex PT-symmetry scattering potentials, we propose that \(R_{left}(-k)=R_{right}(k)\) and \(T(-k)=T(k)\).So far the spectral singularities (SS) of a one-dimensional non-Hermitian scattering potential are witnessed/conjectured to be at most one. We present a new non-Hermitian parametrization of Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies (\(E_*=\alpha^2,\beta^2\)) either in \(T(k)\) or in \(T(-k)\), when \(\alpha\beta>0\). Thirdly, when \(\alpha \beta <0\) it possesses one SS in \(T(k)\) and the other in \(T(-k)\). Lastly, when the potential becomes PT-symmetric \([(\alpha+\beta)=0]\), we get \(T(k)=T(-k)\), it possesses a unique SS at \(E=\alpha^2\) in both \(T(-k)\) and \(T(k)\).

http://arxiv.org/abs/1110.4485

Quantum Physics (quant-ph); Mathematical Physics (math-ph)